Theorem sbhypf 2822

Description: Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf . (Contributed by Raph Levien, 10-Apr-2004.)

Hypotheses

Ref	Expression
sbhypf.1	|- F/ x ps
sbhypf.2	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
sbhypf	|- ( y = A -> ( [ y / x ] ph <-> ps ) )

Distinct variable groups:   x, A   x, y

Allowed substitution hints:   ph( x, y)   ps( x, y)   A( y)

Proof of Theorem sbhypf

Step	Hyp	Ref	Expression
1		vex 2775	. . 3 |- y e. _V
2		eqeq1 2212	. . 3 |- ( x = y -> ( x = A <-> y = A ) )
3	1, 2	ceqsexv 2811	. 2 |- ( E. x ( x = y /\ x = A ) <-> y = A )
4		nfs1v 1967	. . . 4 |- F/ x [ y / x ] ph
5		sbhypf.1	. . . 4 |- F/ x ps
6	4, 5	nfbi 1612	. . 3 |- F/ x ( [ y / x ] ph <-> ps )
7		sbequ12 1794	. . . . 5 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
8	7	bicomd 141	. . . 4 |- ( x = y -> ( [ y / x ] ph <-> ph ) )
9		sbhypf.2	. . . 4 |- ( x = A -> ( ph <-> ps ) )
10	8, 9	sylan9bb 462	. . 3 |- ( ( x = y /\ x = A ) -> ( [ y / x ] ph <-> ps ) )
11	6, 10	exlimi 1617	. 2 |- ( E. x ( x = y /\ x = A ) -> ( [ y / x ] ph <-> ps ) )
12	3, 11	sylbir 135	1 |- ( y = A -> ( [ y / x ] ph <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   F/wnf 1483   E.wex 1515   [wsb 1785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-v 2774

This theorem is referenced by:  mob2  2953  cbvmptf  4138  tfisi  4635  ralxpf  4824  rexxpf  4825  nn0ind-raph  9490